A commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen

This is a mathematical commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces) [24], (1943). This paper is among the last (and may be the last one) that Teichmüller wrote on the theory of moduli. It contains the proof of the so-called Teichmüller existence theorem for a closed surface of genus $g ≥ 2$. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension $6g−6$. Using Brouwer’s theorem of invariance of domain, he shows that this map is a home- omorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar ́e and Klein’s use of the so-called “continuity principle” in their attempts to prove the uniformization theorem.